SECTOR. 



from 60 to 60 b? equal to the radius of the given circle ; 

 then take the extent of the chord of the number of degrees, 

 on each leg of the feAor, and lay it off on the circumference 

 of the given circle. By this ufe, may any regular polygon 

 be infcribed in a given circle, as well as by the line of poly- 

 gons : e. g. in a circle whofe diameter is given to defcribe a 

 regular polygon of 24 fides. Make the given diameter a 

 tranfverfe dillance from 60 to 60 on the fcales of chords ; 

 divide 360 by 24, and take the tranfverfe diftance of 15 and 

 15, the quotient, and this will be the chord of the twenty- 

 fourth part of the circumference. In order to prevent errors, 

 where the diftance is to be repeated feveral times, it will be 

 beft to proceed thus : with the chord of 60 degrees divide 

 the circumference into fix equal parts ; in every divifion of 60 

 degrees lay down, firft, the chord of 15 degrees, and next 

 the chord of 30 degrees, and then the chord of 45 degrees, 

 beginning always at the fame point. Thus the error in 

 taking diftances will not be multiplied into any of the divi- 

 fions following the firft. 



Sector, Ufe of the Line of Polygons on the. I. In a given 

 circle to infcribe a regular polygon, e. g. an oftagon. Open 

 the legs of the feftor, till the tranfverfe diftance of 6 and 6 

 be equal to the given diameter, then will the tranfverfe dif- 

 tance of 8 and 8 be the fide of an oftagon, which may be 

 infcribed in the given circle. In like manner may any other 

 polygon, the number of whofe fides does not exceed 12, be 

 infcribed in a given circle. 



2. On a given line to defcribe a regular polygon, e. g. a 

 pentagon. Make the given line a tranfverfe diftance to 5 

 and 5 : at that opening of the feftor, take the tranfverfe 

 diftance of 6 and 6 ; and with this radius, on the extremi- 

 ties of the line, as centres, defcribe arcs interfering each 

 other ; and on the point of interfcftion, as a centre, with 

 the fame radius, defcribe a circumference palling through 

 the extremities of the given line ; and in this circle may the 

 pentagon, whofe fide is given, be infcribed. By a like pro- 

 cefs may any other polygon, of not more than 12 fides, be 

 defcribed on a given line. 



3. On a right line, to defcribe an ifofceles triangle, hav- 

 ing the angles at the bafe double that at the vertex. Open 

 the feftor till the ends of the given line fall on 10 and 10 on 

 each leg : then take the di'lance from 6 to 6 ; this will be 

 the length of the two equal fides of the triangle. 



Sector, Ufe of the Scales of Sines, Tangents, and Secants 

 on the. By the feveral lines difpofed on the feftor, we have 

 fcales to feveral radiufes : fo that, i, having a length, or 

 radius, given, not exceeding the length of the feftor when 

 opened, we find the chord, fine, &c. thereto: e.g. fuppofe 

 the chord, fine, or tangent, of 10 degrees to a radius of three 

 inches required. Make three inches the aperture, or tranf- 

 verfe diftance, between 60 and 60 on the fcales of chords 

 of the two legs ; then will the fame extent reach from 45 

 to 45 on the fcale of tangents, and from 90 to 90 on the 

 fcale of fines on the other fide : fo that to whatever radius 

 the line of chords is fet, to the fame are all the others fet. 

 In this difpofition, therefore, if the aperture, or tranfverfe 

 diftance, between 10 and ic, on the fcales of chords, be 

 taken with the compafies, it will give the chord of 10 de- 

 grees ; if the tranfverfe diftance of 10 and 10 be in like 

 manner taken, on the fcales of fine?, it will be the fine of 

 10 degrees: laltly, if the tranfverfe diftance of 10 and 10 

 be in like manner taken on the fcales of tangents, it gives 

 the tangent of 10 degrees to the fame radius. 



2. If the chord, or tangent, of 70 degrees were required, 

 for the chord, the tranfverfe diftance of half the arc, viz. 

 35, muft be taken, as before ; which diftance, being re- 

 peated twice, gives the chord of 70 degrees. To find the 



tangent of 70 degrees, to the fame radius, {he fcale of 

 upper tangents muft be ufcd, the other only reaching to 45 : 

 making, therefore, three inchc'; the tranfverfe diftance between 

 45 and 4) at tlie beginning of that fcale ; the extent between 

 70 and 70 degrees, on the fame, will be the tangent of 70 

 degrees to tlirec inches radius. 



3. To find the fecant of an arc, make the given radiu* 

 the tranfverfe diftnnce between o and o on the line of fe- 

 cants ; then will the tranfverfe diftance of 10 and 10, or 70 

 and 70, on tlie faid lines, give the fecant of 10 degrees, or 

 70 degrees. 



The fcales of upper tangents and fecants do not run 

 quite to 76 degrees ; but tiiofe of a greater number of de- 

 grees may be found by tlie feftor in the following manner. 

 Thus, the tangent of any number of degrees may be taken 

 from the feftor at once ; if the radius of the circle can be 

 made a tranfverfe diftance to the complement of thofe de- 

 grees on the lower tangent. H.g. To find the tangent of 78 

 degrees to a radius of two inches. Make two inclies a tranf- 

 verfe diftance of 12 degrees on the lower tangents ; then the 

 tranfverfe diftance of 45 degrees will be the tangent of 78 de- 

 grees. In like manner the fecant of any number of degrees 

 may be taken from the fines, if the radius of the circle can 

 be made a tranfverfe diftance to the cofine of thofe degrees. 

 Thus, making two inches a tranfverfe diftance to the fine of 

 12 degrees, then the tranfverfe diftance of 90 and 90 will be 

 the fecant of 78 degrees. Hence it will be eafy to find tlie 

 degrees anfwering to a given line, cxprefling the length of 

 a tangent or fecant, which is too long to be meafured on 

 thofe fcales, when the feftor is fet to the given radius. 

 Thus, for a tangent, make the given line a tranfverfe dif- 

 tance to 45 and 45 on the lower tangents ; then take the 

 given radius, and apply it to the lower tangents : and the 

 degrees, where it becomes a tranfverfe diftance, give the co- 

 tangent of the degrees anfwering to the given line. And for 

 a fecant, make the given line a tranfverfe diftance to 90 and 

 90 on the fines : then the degrees anfwering to the given 

 radius, applied as a tranfverfe diftance on the fines, will 

 be the cofine of the degrees anfwering to the given fecant 

 line. 



4. If the converfe of any of thefe things were required, 

 that is, if the radius be required, to which a given line is 

 the fine, tangent, or fecant ; it is but making the given line, 

 if a chord, the tranfverfe diftance on the line of chords, 

 between 10 and 10, and then the feftor will ftand at the 

 radius required ; that is, the aperture between 60 and 60, on 

 the faid line, is the radius. 



If the given line were a fine, tangent, or fecant, it is but 

 making it the tranfverfe diftance of the given number of 

 degrees ; then will the diftance of 90 and 90 on the fines, of 

 45 and 45 on the lower tangents near the end of the feftor, 

 and of 45 and 45 on the upper tangents towards the centre 

 of the feftor, and of o and o on the fecants, be the radius. 



J. If the radius, and any line reprefenting a fine, tangent, 

 or fecant, be given, the degrees c.^rrefponding to that line 

 may be found by fetting the feftor to the given radius, ac- 

 cording as a fine, tangent, or fecant, is concerned ; taking 

 the given line betvrccn the compafies, applying the two feet 

 tranfverfely to the fcale concerned, and fliding the feet along 

 till they both reft on like divifions on both leg? ; and the 

 divifions will fhew the degrees and parts correfponding to 

 the given line. 



For the method of determining the degrees anfwering to 

 any tangent, or fecant, that cannot be thus meafured, fee 

 above. 



6. To find the length of a verfed fine to a given number 

 of degrees, and a given radius. Make the tranfverfe dif- 

 tance 



